Loops in AdS from Conformal Field Theory

TL;DR

This paper introduces new methods to compute loop amplitudes in AdS using conformal field theory crossing equations, enabling systematic calculations of non-planar correlators and extending understanding of quantum effects in holographic theories.

Contribution

It presents two independent approaches—solving crossing equations and Mellin space techniques—for calculating AdS loop amplitudes from CFT data, including new results for  theory.

Findings

Successfully computed one-loop bubble diagrams in  theory.

Reproduced known results and derived new four-point functions in  and + theories.

Demonstrated analytic derivation of anomalous dimensions from Mellin amplitudes with infinite poles.

Abstract

We propose and demonstrate a new use for conformal field theory (CFT) crossing equations in the context of AdS/CFT: the computation of loop amplitudes in AdS, dual to non-planar correlators in holographic CFTs. Loops in AdS are largely unexplored, mostly due to technical difficulties in direct calculations. We revisit this problem, and the dual $1/N$ expansion of CFTs, in two independent ways. The first is to show how to explicitly solve the crossing equations to the first subleading order in $1/N^2$, given a leading order solution. This is done as a systematic expansion in inverse powers of the spin, to all orders. These expansions can be resummed, leading to the CFT data for finite values of the spin. Our second approach involves Mellin space. We show how the polar part of the four-point, loop-level Mellin amplitudes can be fully reconstructed from the leading-order data. The anomalous dimensions computed with both methods agree. In the case of $\phi^4$ theory in AdS, our crossing solution reproduces a previous computation of the one-loop bubble diagram. We can go further, deriving part of the four-point function in $\phi^3+\phi^4$ theory in AdS which had never been computed. In the process, we show how to analytically derive anomalous dimensions from Mellin amplitudes with an infinite series of poles, and discuss applications to more complicated cases such as the ${\cal N}=4$ super-Yang-Mills theory.